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For example, what is the first Chern class of $X:=\mathbb{P}(T\mathbb{P}^3)$ and $Y:=\mathbb{P}(T^*\mathbb{P}^3)$?

I am asking this question because I saw an essay today by F.Hirzebruch, saying that the number $c_1^5$ of $X$ is 4500, while that of $Y$ is 4860, thus providing an example of two diffeomorphic manifolds with different Chern numbers. He did not give the details, and that's why I am here for help.

Hopefully this is not a too stupid question, and thank you all in advance.

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Try using the Euler sequence associated to a projective bundle (see mathoverflow.net/questions) in order to do computations. It should not be difficult, but a bit long. I have no time now but maybe tomorrow I can try, if no one answered in the meantime. –  Francesco Polizzi Oct 23 '11 at 17:56
    
the correct link is mathoverflow.net/questions/36658/… –  Francesco Polizzi Oct 23 '11 at 18:04
    
IIRC you can find this (or a similar example) in Hirzebruch's paper with Borel (Characteristic classes and homogeneous spaces, II). The details might involve some Lie theory. –  Faisal Oct 23 '11 at 18:05
    
Yes -- see the example at the bottom of page 340. (The paper is on JSTOR: jstor.org/pss/2372747) –  Faisal Oct 23 '11 at 18:11

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